A A Survey of Qualitative Spatial and Temporal Calculi — Algebraic and Computational Properties

partition schemes. Ligozat and Renz [2004] note that most spatial and temporal calculi are based on a set of JEPD (jointly exhaustive and pairwise disjoint) domain relations. The following definition is predominant in the QSTR literature [Ligozat and Renz 2004; Cohn and Renz 2008]. Definition 3.1. Let U be a universe and R a set of non-empty domain relations of the same arity n. R is called a set of JEPD relations over U if the relations in R are jointly exhaustive, i.e., U = ⋃ r∈R r, and pairwise disjoint. An n-ary abstract partition scheme is a pair (U ,R) whereR is a set of JEPD relations over the universe U . The relations in R are called base relations. ACM Computing Surveys, Vol. V, No. N, Article A, Publication date: January YYYY. A Survey of Qualitative Spatial and Temporal Calculi A:7 Ex. A.4 In Definition 3.1, the universe U represents the set of all (spatial or temporal) entities. The main ingredients of a calculus will be relation symbols representing the base relations in the underlying partition scheme. A constraint linking an n-tuple t of entities via a relation symbol will thus represent complete information (modulo the qualitative abstraction underlying the partition scheme) about t. Incomplete information is modeled by t being in a composite relation, which is a set of relation symbols representing the union of the corresponding base relations. The set of all relation symbols represents the universal relation (the union of all base relations) and indicates that no information is available. Ex. A.5 The requirement that all base relations are JEPD ensures that every n-tuple of entities belongs to exactly one base relation. Thanks to PD (pairwise disjointness), there is a unique way to represent any composite relation using relation symbols and, due to JE (joint exhaustiveness), the empty relation can never occur in a consistent set of constraints, which is relevant for reasoning, see Section 3.2. Ex. A.6 Partition schemes, identity, and converse. Ligozat and Renz [2004] base their definition of a (binary) qualitative calculus on the notion of a partition scheme, which imposes additional requirements on an abstract partition scheme. In particular, it requires that the set of base relations contains the identity relation and is closed under the converse operation. The analogous definition by Condotta et al. [2006] captures relations of arbitrary arity. Before we define the notion of a partition scheme, we discuss the generalization of identity and converse to the n-ary case. The binary identity relation is given as usual by id = {(u, u) | u ∈ U}. (1) Ex. A.7 The most inclusive way to generalize (1) to the n-ary case is to fix a setM of numbers of all positions where tuples in id are required to agree. Thus, an n-ary identity relation is a domain relation idnM with M ⊆ {1, . . . , n} and |M | > 2, which is defined by idnM = {(u1, . . . , un) ∈ U | ui = uj for all i, j ∈M}. This definition subsumes the “diagonal elements” ∆ij of Condotta et al. [2006] for the case |M | = 2. However, it is not enough to restrict attention to |M | = 2 because there are ternary calculi which contain all identities id31,2, id 3 1,3, id 3 2,3, and id 3 1,2,3, an example being the LR calculus, which was described as “the finest of its class” [Scivos and Nebel 2005]. Since the relations in an n-ary abstract partition scheme are JEPD, all identities idnM are either base relations or subsumed by those. The stronger notion of a partition scheme should thus require that all identities be made explicit. For binary relations, id from (1) is the unique identity relation id2{1,2}. The standard definition for the converse operation ̆ on binary relations is r̆ = {(v, u) | (u, v) ∈ r}. (2) Ex. A.8 In order to generalize the reversal of the pairs (u, v) in (2) to n-ary tuples, we consider arbitrary permutations of n-tuples. An n-ary permutation is a bijection π : {1, . . . , n} → {1, . . . , n}. We use the notation π : (1, . . . , n) 7→ (i1, . . . , in) as an abbreviation for “π(1) = i1, . . . , π(n) = in”. The identity permutation ι : (1, . . . , n) 7→ (1, . . . , n) is called trivial; all other permutations are nontrivial. ACM Computing Surveys, Vol. V, No. N, Article A, Publication date: January YYYY. A:8 Frank Dylla et al. A finite set P of n-ary permutations is called generating if each n-ary permutation is a composition of permutations from P . For example, the following two permutations form a (minimal) generating set: sc : (1, . . . , n) 7→ (2, . . . , n, 1) (shortcut) hm : (1, . . . , n) 7→ (1, . . . , n− 2, n, n− 1) (homing) The names have been introduced in Freksa and Zimmermann [1992] for ternary permutations, together with a name for a third distinguished permutation: inv : (1, . . . , n) 7→ (2, 1, 3 . . . , n) (inversion) Condotta et al. [2006] call shortcut “rotation” (ry) and homing “permutation” (r#). Ex. A.9 For n = 2, sc, hm and inv coincide; indeed, there is a unique minimal generating set, which consists of the single permutation ̆ : (1, 2) 7→ (2, 1). For n > 3, there are several generating sets, e.g., {sc,hm} and {sc, inv}. Now an n-ary permutation operation is a map · that assigns to each n-ary domain relation r an n-ary domain relation denoted by r, where π is an n-ary permutation and the following holds: r = {(uπ(1), . . . , uπ(n)) | (u1, . . . , un) ∈ r} We are now ready to give our definition of a partition scheme, lifting Ligozat and Renz’s binary version to the n-ary case, and generalizing Condotta et al.’s n-ary version to arbitrary generating sets. Definition 3.2. An n-ary partition scheme (U ,R) is an n-ary abstract partition scheme with the following two additional properties. (1) R contains all identity relations idnM , M ⊆ {1, . . . , n}, |M | > 2. (2) There is a generating set P of permutations such that, for every r ∈ R and every π ∈ P , there is some s ∈ R with r = s. Ob. B.1 Ex. A.10, A.11 It is important to note that violations of Definition 3.2 (e.g., depicted in Example A.11) are not necessarily bugs in the design of the respective calculi – in fact they are often a feature of the corresponding representation language, which is deliberately designed to be just as granular as necessary, and may thus omit some identity relations or converses/compositions of base relations. Ex. A.12 Thus violations of Definition 3.2 are unavoidable, and we adopt the more general notion of an abstract partition scheme. Calculi. Intuitively, a qualitative (spatial or temporal) calculus is a symbolic representation of an abstract partition scheme and additionally represents the composition operation on the relations involved. As before, we need to discuss the generalization of binary composition to the n-ary case before we can define it precisely. For binary domain relations, the standard definition of composition is: r ◦ s = {(u,w) | ∃v ∈ U : (u, v) ∈ r and (v, w) ∈ s} (3) Ex. A.13 We are aware of three ways to generalize (3) to higher arities. The first is a binary operation on the ternary relations of the calculus double-cross (2-cross) [Freksa 1992b; Freksa and Zimmermann 1992] (see also Fig. 8 in the appendix): r ◦FZ s = {(u, v, w) | ∃x : (u, v, x) ∈ r and (v, x, w) ∈ s} Ex. A.14 ACM Computing Surveys, Vol. V, No. N, Article A, Publication date: January YYYY. A Survey of Qualitative Spatial and Temporal Calculi A:9 A second alternative results in n(n − 1) binary operations i◦j [Isli and Cohn 2000; Scivos and Nebel 2005]: the composition of r and s consists of those n-tuples that belong to r (respectively, s) if the i-th (respectively, j-th) component is replaced by some uniform element v. r i◦j s = {(u1, . . . , un) | ∃v : (u1, . . . , ui−1, v, ui+1, . . . , un) ∈ r and (u1, . . . , uj−1, v, uj+1, . . . , un) ∈ s } In the ternary case, this yields, for example: r 3◦2 s = {(u, v, w) | ∃x : (u, v, x) ∈ r and (u, x, w) ∈ s} (4) If we assume, for example, that the underlying partition scheme speaks about the relative position of points, we can consider (4) to say: if the position of x relative to u and v is determined by the relation r (as given by (u, v, x) ∈ r) and the position of w relative to u and x is determined by the relation s (as given by (u, x, w) ∈ s), then the position of w relative to u and v can be inferred to be determined by r 3◦2 s. The third is perhaps the most general, resulting in an n-ary operation [Condotta et al. 2006]: ◦(r1, . . . , rn) consists of those n-tuples which, for every i = 1, . . . , n, belong to the relation ri whenever their i-th component is replaced by some uniform v. ◦(r1, . . . , rn) = {(u1, . . . , un) | ∃v ∈ U : (u1, . . . , un−1, v) ∈ r1 and (u1, . . . , un−2, v, un) ∈ r2 and . . . and (v, u2, . . . , un) ∈ rn} (5) Ex. A.15 For binary domain relations, all these alternative approaches collapse to (3). In the light of the diverse views on composition, we define a composition operation on n-ary domain relations to be an operation of arity 2 6 m 6 n on n-ary domain relations, without imposing additional requirements. Those are not necessary for the following definitions, which are independent of the particular choice of composition. We now define our minimal notion of a calculus, which provides a set of symbols for the relations in an abstract partition scheme (Rel), and for some choice of nontrivial permutation operations (̆ , . . . , ̆) and some composition operation ( ). Definition 3.3. An n-ary qualitative calculus is a tuple (Rel, Int, ̆, . . . , ̆, ) with k > 1 and the following properties. — Rel is a finite, non-empty set of n-ary relation symbols (denoted r, s, t, . . . ). The subsets of Rel, including singletons, are called composite relations (denoted R,S, T, . . . ). — Int = (U , φ, ·1 , . . . , ·k , ◦) is an interpretation with the following properties. — U is a universe. — φ : Rel → 2Un is an injective map assigning an n-ary relation over U to each relation symbol, such that (U , {φ(r) | r ∈ Rel}) is an abstract partition scheme. The map φ is extended to composite relations R ⊆ Rel by setting φ(R) = ⋃ r∈R φ(r). — {·1 , . . . , ·k} is a set of n-ary nontrivial permutation operations. — ◦ is a composition operation on n-ary domain relations that has arity 2 6 m 6 n. — Every permutation operation ̆ is a map ̆ : Rel→ 2Rel that satisfies φ(r̆ ) ⊇ φ(r)i (6) for every r ∈ Rel. The operation ̆ is extended to composite relations R ⊆ Rel by setting R ̆ = ⋃ r∈R r̆ . — The composition operation is a map : Rel → 2Rel that satisfies φ( (r1, . . . , rm)) ⊇ ◦(φ(r1), . . . , φ(rm)) (7) ACM Computing Surveys, Vol. V, No. N, Article A, Publication date: January YYYY. A:10 Frank Dylla et al. for all r1, . . . , rm ∈ Rel. The operation is extended to composite relations R1, . . . , Rm ⊆ Rel by setting (R1, . . . , Rm) = ⋃ r1∈R1 · · · ⋃ rm∈Rm (r1, . . . , rm). In the special case of binary relations, the natural converse is the only non-trivial permutation operation, i.e., k = 1. Ob. B.2 Due to the last sentence of Definition 3.3, the composition operation of a calculus is uniquely determined by the composition of each pair of relation symbols. This information is usually stored in an m-dimensional table, the composition table. Ex. A.16, A.17, A.18 Abstract versus weak and strong operations. We call permutation and composition operations with Properties (6) and (7) abstract permutation and abstract composition, following Ligozat’s naming in the binary case [Ligozat 2005]. For reasons explained further below, our notion of a qualitative calculus imposes weaker requirements on the permutation operation than Ligozat and Renz’s notions of a weak (binary) representation [Ligozat 2005; Ligozat and Renz 2004] or the notion of a (binary) constraint algebra [Nebel and Scivos 2002]. The following definition specifies those stronger variants, see, e.g., Ligozat and Renz [2004].versus weak and strong operations. We call permutation and composition operations with Properties (6) and (7) abstract permutation and abstract composition, following Ligozat’s naming in the binary case [Ligozat 2005]. For reasons explained further below, our notion of a qualitative calculus imposes weaker requirements on the permutation operation than Ligozat and Renz’s notions of a weak (binary) representation [Ligozat 2005; Ligozat and Renz 2004] or the notion of a (binary) constraint algebra [Nebel and Scivos 2002]. The following definition specifies those stronger variants, see, e.g., Ligozat and Renz [2004]. Definition 3.4. Let (Rel, Int, ̆, . . . , ̆, ) be a qualitative calculus based on the interpretation Int = (U , φ, ·1 , . . . , ·k , ◦). The permutation operation ̆ is a weak permutation if, for all r ∈ Rel: r̆ i = ⋂ {S ⊆ Rel | φ(S) ⊇ φ(r)i} (8) The permutation operation ̆ is a strong permutation if, for all r ∈ Rel: φ(r̆ ) = φ(r)i (9) The composition operation is a weak composition if, for all r1, . . . , rm ∈ Rel: (r1, . . . , rm) = ⋂ {S ⊆ Rel | φ(S) ⊇ ◦(φ(r1), . . . , φ(rm))} (10) The composition is a strong composition if, for all r1, . . . , rm ∈ Rel: φ( (r1, . . . , rm)) = ◦(φ(r1), . . . , φ(rm)) (11) In the literature, the equivalent variant r̆ i = {s ∈ Rel | φ(s) ∩ φ(r)i 6= ∅} of Equation (8) is sometimes found; analogously for Equation (10). Ex. A.19, A.20 In terms of composition tables, abstract composition requires that each cell corresponding to (r1, . . . , rm) contains at least those relation symbols t whose interpretation intersects with ◦(φ(r1), . . . , φ(rm)). Weak composition additionally requires that each cell contains exactly those t. Strong composition, in contrast, imposes a requirement on the underlying partition scheme: whenever φ(t) intersects with ◦(φ(r1), . . . , φ(rm)), it has to be contained in ◦(φ(r1), . . . , φ(rm)). Analogously for permutation. The above “at least” is a crucial requirement: if some cell did not contain any relation symbol t as above, then the composition table would give rise to unsound inferences, (e.g., described in Example A.20). Abstractness as in Properties (6) and (7) thus captures minimal requirements to the operations in a qualitative calculus that ensure soundness of reasoning, as described in Section 3.2. Along the same lines, adding unnecessary relations to a cell in the table leads to weaker inferences and thus amounts to a loss of knowledge. Weakness (Properties (8) and (10)) ensures that this loss is kept to the unavoidable minimum. This last observation is presumably the reason why existing calculi (see Section 3.4) typically have at least weak operations – we are not aware of any calculus with only abstract operations. ACM Computing Surveys, Vol. V, No. N, Article A, Publication date: January YYYY. A Survey of Qualitative Spatial and Temporal Calculi A:11 In Section 3.2, we will see that abstract composition is a minimal requirement for ensuring soundness of the most common reasoning algorithm, a-closure, and review the impact of the various strengths of the operations on reasoning algorithms. The three notions form a hierarchy: FACT 3.5. Every strong permutation (composition) is weak, and every weak permutation (composition) is abstract. C.1 It suffices to postulate the properties weakness and strongness with respect to relation symbols only: they carry over to composite relations as shown in Fact 3.6. FACT 3.6. Given a qualitative calculus (Rel, Int, ̆, . . . , ̆, ) the following holds. For all composite relations R ⊆ Rel and i = 1, . . . , k: